Assertion (A) : If A is square matrix such that ` A ^(2) = A, ` then `( I + A)^(2) - 3 A = I`
Reason (R) : AI = IA = A

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is true

Answer

To solve the problem, we need to verify the assertion and reason provided in the question.

### Step-by-Step Solution

1. **Given Information**:
We have a square matrix \( A \) such that \( A^2 = A \). This means \( A \) is an idempotent matrix.

2. **Expression to Prove**:
We need to show that \( (I + A)^2 - 3A = I \).

3. **Expand the Left-Hand Side**:
We start by expanding \( (I + A)^2 \):
\[ (I + A)^2 = I^2 + 2IA + A^2 \]
Using the properties of matrices, we know \( I^2 = I \) and \( A^2 = A \). Also, \( IA = A \) and \( AI = A \). Thus, we can rewrite the expression:
\[ (I + A)^2 = I + 2A + A \]
Simplifying this gives:
\[ (I + A)^2 = I + 3A \]

4. **Substituting Back into the Expression**:
Now we substitute this back into our original expression:
\[ (I + A)^2 - 3A = (I + 3A) - 3A \]

5. **Simplifying the Expression**:
This simplifies to:
\[ (I + 3A) - 3A = I \]

6. **Conclusion**:
Therefore, we have shown that:
\[ (I + A)^2 - 3A = I \]
This confirms the assertion is true.

7. **Reason Verification**:
The reason given is \( AI = IA = A \). This is a fundamental property of matrices, where multiplying any matrix by the identity matrix results in the original matrix. This property is essential for the calculations we performed.

### Final Result
Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion.